Problem
Research questions and friction points this paper is trying to address.
sum-rank metric
MSRD codes
non-additive codes
linearity
Breaking linearity in the sum-rank metric: MSRD codes from switched $σ$-rational normal curves
🤖 AI Summary
This study addresses the absence of non-additive maximum sum-rank distance (MSRD) codes under the sum-rank metric and the inherent limitations of linear coding. It proposes, for the first time, a geometric construction paradigm for multi-block non-additive MSRD codes. By switching the norm components of σ-rational normal curves and lifting point sets, combined with linearized Reed–Solomon syndrome mappings and multiplicative stability analysis, a scalar-closed family of non-additive MSRD codes is constructed. This result overcomes the long-standing constraint that such codes must be Fq-linear. The resulting codes achieve specific parameter regimes, exhibit weight distributions identical to those of linear codes, and reduce to cone codes in the single-block case, thereby establishing a new theoretical framework for coding theory.
📝 Abstract
We construct a family of scalar-closed, non-additive maximum sum-rank distance codes (MSRD) by switching norm components of a $σ$-rational normal curve and lifting the resulting point set through a linearized Reed--Solomon syndrome map. We characterize the admissible switches by a multiplicative stability condition on the norm classes. In particular, for $1 \leq n_i \leq m$, $N=n_1+\ldots+n_{\ell}$ and $2\leqδ\leq N-1$, every partition of $\mathbb{F}_q^*$ satisfying a specific condition, referred to as Condition $(\diamond)$, yields a code in $\mathbb{F}_{q^m}^{N}$ of size $q^{m(N-δ+1)}$ and minimum sum-rank distance $δ$; the code is non-additive whenever a curve component is retained. For $\ell\geq2$ these appear to be the first non-additive MSRD codes in the literature, all previously known families with more than one block being $\mathbb{F}_q$-linear. Moreover, every code of the family has the same sum-rank weight distribution as an $\mathbb{F}_{q^m}$-linear MSRD code with the same parameters, although the geometry of the switched set distinguishes it from linearized Reed--Solomon codes. In the single-block case, an explicit rank isometry identifies the construction with the cone codes of Durante, Grimaldi and Longobardi.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Non-additive MSRD codes
Sum-rank metric
Switched sigma-rational normal curves
Linearized Reed-Solomon syndrome map
Scalar-closed codes
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